because symbol — Definition, Formula & Examples
The because symbol (∵) is three dots arranged in a triangle with one dot on top and two on the bottom, used in mathematical writing to mean 'because' or 'since.' It introduces a reason or given fact that supports a conclusion.
The symbol ∵ is a logical notation placed before a premise or justification in a proof or argument, indicating that the statement it precedes serves as the basis or reason for a subsequent conclusion, often marked by the therefore symbol ∴.
How It Works
Place ∵ before the statement that gives your reason, and ∴ before the statement that follows from it. Think of ∵ as answering 'why?' and ∴ as answering 'so what follows?' The two symbols are companions: ∵ introduces the cause, ∴ introduces the effect. In a proof, you might chain several ∵ statements before arriving at a single ∴ conclusion.
Example
Problem: Use the because and therefore symbols to express: 'x equals 6 because 2x equals 12.'
State the reason: Write the known fact preceded by the because symbol.
State the conclusion: Divide both sides by 2 and write the result preceded by the therefore symbol.
Answer: ∵ 2x = 12, ∴ x = 6
Why It Matters
Geometry proofs and formal logic courses use ∵ and ∴ as shorthand to keep reasoning compact and readable. Recognizing these symbols lets you follow two-column proofs and mathematical arguments without stumbling over notation.
Common Mistakes
Mistake: Confusing ∵ (because, dot on top) with ∴ (therefore, dot on bottom).
Correction: Remember: ∵ has the single dot on top like a finger pointing up to a reason above, while ∴ has the single dot on the bottom like an arrow pointing down to a conclusion below.